All seems true.
C. Let M be a adjacency matrix for R & M' be for $R^{-1}$ . M & M' are mutually transpose of each other, but diagonal remains same for both. Therefore $R\cap{R^{-1}}$ is nothing but $2^{n}$ numbers of diagonal fills of adjacency matrix which is sign of anti symmetric relations.
Let me know where I'm wrong if I'm.